Question
Answer
$$(x-4)(x^2+1)(2x-3)(-x+2)=-2x^5+15x^4-36x^3+39x^2-34x+24$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(x-4)(x^2+1)(2x-3)(-x+2)& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}(x^3+x-4x^2-4)(2x-3)(-x+2) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}(2x^4-11x^3+14x^2-11x+12)(-x+2) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}-2x^5+15x^4-36x^3+39x^2-34x+24\end{aligned} $$ | |
| ① | Multiply each term of $ \left( \color{blue}{x-4}\right) $ by each term in $ \left( x^2+1\right) $. $$ \left( \color{blue}{x-4}\right) \cdot \left( x^2+1\right) = x^3+x-4x^2-4 $$ |
| ② | Multiply each term of $ \left( \color{blue}{x^3+x-4x^2-4}\right) $ by each term in $ \left( 2x-3\right) $. $$ \left( \color{blue}{x^3+x-4x^2-4}\right) \cdot \left( 2x-3\right) = 2x^4-3x^3+2x^2-3x-8x^3+12x^2-8x+12 $$ |
| ③ | Combine like terms: $$ 2x^4 \color{blue}{-3x^3} + \color{red}{2x^2} \color{green}{-3x} \color{blue}{-8x^3} + \color{red}{12x^2} \color{green}{-8x} +12 = \\ = 2x^4 \color{blue}{-11x^3} + \color{red}{14x^2} \color{green}{-11x} +12 $$ |
| ④ | Multiply each term of $ \left( \color{blue}{2x^4-11x^3+14x^2-11x+12}\right) $ by each term in $ \left( -x+2\right) $. $$ \left( \color{blue}{2x^4-11x^3+14x^2-11x+12}\right) \cdot \left( -x+2\right) = \\ = -2x^5+4x^4+11x^4-22x^3-14x^3+28x^2+11x^2-22x-12x+24 $$ |
| ⑤ | Combine like terms: $$ -2x^5+ \color{blue}{4x^4} + \color{blue}{11x^4} \color{red}{-22x^3} \color{red}{-14x^3} + \color{green}{28x^2} + \color{green}{11x^2} \color{orange}{-22x} \color{orange}{-12x} +24 = \\ = -2x^5+ \color{blue}{15x^4} \color{red}{-36x^3} + \color{green}{39x^2} \color{orange}{-34x} +24 $$ |
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